🧮 algebra

1. **Prime Factorization and Multiplication** - Problem: Verify the prime factorization and multiplication of $2^4 \times 3^6 \times 5^2 \times 7^4$.

1. The problem is to evaluate the function $f(x)$ at $x=3$ and verify if $f(3)=4$ or $f(3)=3$. 2. To solve this, we need the explicit formula or rule for the function $f(x)$, which

1. **State the problem:** Simplify the expression $2(a^2 + b^2)$. 2. **Formula and rules:** The distributive property states that $k(x + y) = kx + ky$, where $k$, $x$, and $y$ are

1. Problem (14a): Given the function $f(x) = x^4 + p_1$ defined on $\mathbb{R}$ with an inverse function bounded on $\mathbb{R}$, find the values of $p_1$ and $p_2$. 2. To have an

1. **State the problem:** We need to calculate $8 \div 2^{4/5}$ and express the answer as a mixed number in its lowest terms. 2. **Recall the rules:** Division by an exponent means

1. **State the problem:** Simplify the expression $2(a^2 + b^2)$. 2. **Formula and rules:** The distributive property states that $k(x + y) = kx + ky$. Here, $k=2$, $x=a^2$, and $y

1. **State the problem:** Solve the cubic equation $$x^3 + 3x^2 + 3x + 9 = 0$$ for the values of $x$. 2. **Recognize the form:** The polynomial resembles the expansion of a cube of

1. **State the problem:** Solve the equation $$3x^4 + 18x^2 - 48 = 0$$ for $x$. 2. **Identify the type of equation:** This is a quartic equation but can be treated as a quadratic i

1. **State the problem:** We are given two equations: $$a + b = 5$$

1. **Problem Statement:** Arrange the given slope values $3$, $\frac{4}{5}$, $-4$, $-\frac{11}{2}$, $1.5$, and $0$ in order from least steep to most steep. 2. **Understanding Slope

1. **Problem Statement:** We are given the linear function $f(x) = -4x$ and need to understand its properties and graph. 2. **Formula and Explanation:** A linear function has the f

1. Let's solve a simple algebra problem: Find $x$ if $2x + 3 = 7$. 2. The formula used here is to isolate $x$ by performing inverse operations.

1. Задача: Знайти область визначення функції $y = \sqrt{\frac{3 - x}{4x^2 - 9x - 13}}$. 2. Формула: Область визначення функції підкореневого виразу $\geq 0$ та знаменник не дорівню

1. **State the problem:** Solve the inequality $4 - 7x \geq 2(x + 3)$. 2. **Write the inequality:**

1. **State the problem:** Solve the equation $x - 5 = 6$ for $x$. 2. **Formula and rules:** To solve for $x$, we want to isolate $x$ on one side of the equation. We do this by perf

1. **State the problem:** Solve the quadratic equation $x^2 + 6x - 5 = 0$ for $x$. 2. **Formula used:** The quadratic formula is given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

1. **State the problem:** Solve the quadratic equation $6x^2 + 6x - 5 = 0$ for $x$. 2. **Formula used:** For a quadratic equation $ax^2 + bx + c = 0$, the solutions are given by th

1. **State the problem:** Simplify the expression $$\sqrt[4]{768x^8y^5}$$. 2. **Recall the formula:** For any non-negative real numbers and variables, $$\sqrt[4]{a^b} = a^{\frac{b}

1. **State the problem:** Simplify the expression $$7^{-\frac{5}{6}} \cdot 7^{-\frac{1}{6}}$$. 2. **Recall the exponent multiplication rule:** When multiplying powers with the same

1. **State the problem:** Solve the quadratic equation $x^2 + 6x - 5 = 0$ for $x$. 2. **Formula used:** The quadratic formula is used to solve equations of the form $ax^2 + bx + c

1. **State the problem:** Simplify the expression $$7^{-\frac{5}{6}} \cdot 7^{-\frac{1}{6}}$$ and choose the correct answer from options A, B, C, or D. 2. **Recall the exponent mul